Question

The arithmetic mean of 12 observations is7.5.If the arithmetic mean of 7 of these observations is 6.5,the mean of remaining observations is?

Answer 1

Mean of remaining observations is 8.9.

Step-by-step explanation:

We are given that the arithmetic mean of 12 observations is 7.5. The arithmetic mean of 7 of these observations is 6.5 and we have to find the mean of remaining observations.

Now, the mean of any data is given by the following formula;

               $\text{Mean}= \frac{\text{Sum of all observations}}{\text{Total number of observations}}$

                                   Or

                     $\bar X = \frac{\sum X}{n}$  

So, the arithmetic mean of 12 observations is given as 7.5, i.e;

                      $\bar X = \frac{\sum X_1_2}{n}$

where, $\sum X_1_2$ = Sum of all 12 observations  

                      7.5  =  $\frac{\sum X_1_2}{12}$

So,  $\sum X_1_2 = 7.5 \times 12$  =  90.

Now, also the arithmetic mean of 7 of these observations is 6.5;

                   Mean of 7 observations = $\frac{\sum X_7}{7}$

                                6.5  =  $\frac{\sum X_7}{7}$

where, $\sum X_7$ = Sum of 7 of these observations

So,  $\sum X_7 = 6.5 \times 7$  =  45.5.

Now, the sum of remaining 5 observations is  $\sum X_5$, that is;

                     $\sum X_1_2 = \sum X_7 + \sum X_5$

                      $\sum X_5$ = $\sum X_1_2 -\sum X_7$

                                = 90 - 45.5 = 44.5

Hence, the mean of remaining 5 observations =  $\frac{\sum X_5}{5}$

                                                                              = $\frac{44.5}{5}$ = 8.9

Therefore, the mean of remaining observations is 8.9.